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NEW FOUNDATIONS FOR GEOMETRY-TWO NON-ADDITIVE LANGUAGES By Shai M. J. Haran *VG*
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- Buy it nowNEW FOUNDATIONS FOR GEOMETRY-TWO NON-ADDITIVE LANGUAGES By Shai M. J. Haran *VG*
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Item specifics
- Condition
- Very Good
- Seller notes
- ISBN-10
- 147042312X
- Book Title
- New Foundations for Geometry-two Non-additive Languages for
- Item Height
- 10.25 inches
- ISBN
- 9781470423124
- Subject Area
- Mathematics
- Publication Name
- New Foundations for Geometry: Two Non-Additive Languages for Arithmetical Geometry
- Publisher
- American Mathematical Society
- Item Length
- 10 in
- Subject
- Algebra / General, Geometry / Algebraic
- Publication Year
- 2017
- Series
- Memoirs of the American Mathematical Society Ser.
- Type
- Textbook
- Format
- Trade Paperback
- Language
- English
- Item Width
- 7 in
- Number of Pages
- 202 Pages
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Product Information
We give two simple generalizations of commutative rings. They form (co)-complete categories, that contain commutative (semi-) rings (e.g. with the usual multiplication ). But they also contains the "integers" (and ), and the "residue fields" (and ), of the real (and complex) numbers. Here is the collection of unit balls, and is the collection of spheres augmented with a . The initial object is "the field with one element" .One generalization, - the "commutative generalized rings", is an axiomatization of finitely generated free modules over a commutative ring, together with the operations of multiplication and contraction. This is the more geometric language: for any we associate its (symmetric) spectrum, , a compact Zariski space, with a sheaf of over it. By glueing such spectra we get generalized schemes , a full sub-category of the locally-generalized-ringed-spaces. For a number field , with the ring of integers , the compatification of is a pro-object , and its points are the valuation-sub- of : .For , we have a (co)-complete abelian category of - modules with enough injectives and projectives. For in , we obtain the - module of Kähler differentials , satisfying all the usual properties. We compute the universal derivation .All these remain true for the second generalization - the "commutative with involution", the axiomatization of the category of finitely generated free -modules with -linear maps, and the operations of composition,direct sum, and taking transpose.This is the more "linear", or K-theoretic language: for , we have its algebraic K-theory spectum: , and for we obtain the sphere spectrum .For a compact valuation we associate a "zeta" function, so that we obtain the usual factor for the p-adic integers , while we get for the real integers .For , we define the category of vector bundles over , by a certain completion of the categories of vector bundles on the finite layers . For a number field , the isomorphism classes of rank vector bundles over are in natural bijection withwhere (resp. ) for real (resp. complex) place of . E.g. for : , and for : .We have the following "commutative" diagram of adjunctions:where is the left adjoint of the forgetfull functor and .We describe the ordinary commutative (semi)- ring associated by the right adjoint functor to the - fold tensor product (resp. ).Its elements are (non-uniquely) represented as , where are finite rooted trees, with maps , and is a bijection of their leaves , and for we have in addition signs.
Product Identifiers
Publisher
American Mathematical Society
ISBN-10
147042312x
ISBN-13
9781470423124
eBay Product ID (ePID)
19038520260
Product Key Features
Number of Pages
202 Pages
Language
English
Publication Name
New Foundations for Geometry: Two Non-Additive Languages for Arithmetical Geometry
Publication Year
2017
Subject
Algebra / General, Geometry / Algebraic
Type
Textbook
Subject Area
Mathematics
Series
Memoirs of the American Mathematical Society Ser.
Format
Trade Paperback
Dimensions
Item Length
10 in
Item Width
7 in
Additional Product Features
LCCN
2016-055232
Dewey Edition
23
Series Volume Number
246
Target Audience
Scholarly & Professional
Illustrated
Yes
Dewey Decimal
516.3/5
Lc Classification Number
Qa242.5.H36 2017
Table of Content
Introduction Part I. \mathbb{F}-\mathcal{R}ings: Definition of $\mathbb{F}$-$\mathcal{R}$ings Appendix A Examples of $\mathbb{F}$-$\mathcal{R}$ings Appendix B Geometry Symmetric geometry Pro - limits Vector bundles Modules Part II. Generalized Rings: Generalized Rings Ideals Primes and spectra Localization and sheaves Schemes Products Modules and differentials Appendix C Bibliography
Copyright Date
2017
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